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A Platonic solid is a convex polyhedron whose faces all use the same regular polygon and such that the same number of faces meet at all its vertices. Compare with the Kepler-Poinsot solids, which are not convex, and the Archimedean and Johnson solids, which while made of regular polygons are not themselves regular.
There are five Platonic solids, all known to the ancient Greeks:
| Name and picture | Face polygon | Faces | Edges | Vertices | Faces meeting at each vertex |
Symmetry group |
|---|---|---|---|---|---|---|
| tetrahedron |
triangle | 4 | 6 | 4 | 3 | Td |
| cube (hexahedron) |
square | 6 | 12 | 8 | 3 | Oh |
| octahedron |
triangle | 8 | 12 | 6 | 4 | Oh |
| dodecahedron |
pentagon | 12 | 30 | 20 | 3 | Ih |
| icosahedron |
triangle | 20 | 30 | 12 | 5 | Ih |
| Contents |
That there are only five such three-dimensional solids is easily demonstrated. To create a vertex, at least three faces must meet at a point and the total of their angles must be less than 360°, i.e the corners of the face must be less than 360°/3=120°. The only polygons meeting these requirements are the triangle, square, and pentagon.
Note that if you connect the centers of the faces of a tetrahedron, you get another tetrahedron. If you connect the centers of the faces of an octahedron, you get a cube, and vice versa. If you connect the centers of the faces of a dodecahedron, you get an icosahedron, and vice versa. These pairs are said to be dual polyhedra.
The Platonic solids are named after Plato, who wrote about them in Timaeus. Plato learned about these solids from his friend Theaetetus. The constructions of the solids are included in Book XIII of Euclid's Elements. Proposition 13 describes the construction of the tetrahedron, proposition 14 of the octahedron, proposition 15 of the cube, proposition 16 of the icosahedron, and proposition 17 of the dodecahedron.
Plato conceived the four classical elements as atoms with the geometrical shapes of four of the five platonic solids that had been discovered by the Pythagoreans (in the Timaeus). These are, of course, not the true shapes of atoms; but it turns out that they are some of the true shapes of packed atoms and molecules, namely crystals: The mineral salt sodium chloride occurs in cubic crystals, fluorite (calcium floride) in octahedra, and pyrite in dodecahedra (see uses below).
This concept linked fire with the tetrahedron, earth with the cube, air with the octahedron and water with the icosahedron. There was logical reasoning behind these associations: the heat of fire feels sharp and stabbing (like little tetrahedra). Air is made of the octahedron; its minuscule components are so smooth that one can barely feel it. Water, the icosahedron, flows out of one's hand when picked up, as if it is made of tiny little balls. By contrast, a highly un-spherical solid, the hexahedron (cube) represents earth. These clumsy little solids cause dirt to crumble and breaks when picked up, in stark difference to the smooth flow of water.
The fifth Platonic Solid, the dodecahedron, Plato obscurely remarks, "...the god used for arranging the constellations on the whole heaven" (Timaeus 55). He didn't really know what else to do with it. Aristotle added a fifth element, aithêr (aether in Latin, "ether" in English) and postulated that the heavens were made of this element, but he had no interest in matching it with Plato's fifth solid.
Historically, Johannes Kepler followed the custom of the Renaissance in making mathematical correspondences, and identified the five platonic solids with the five planets – Mercury, Venus, Mars, Jupiter, Saturn which themselves represented the five classical elements.
When the Platonic polyhedra are inscribed in a sphere, they occupy the following percentages of that sphere's volume:
Note that, despite the common assumption, the dodecahedron occupies significantly more of the sphere's volume than the apparently more spherical icosahedron. The corners of the dodecahedron are less sharp than the corners of the icosahedron, and therefore fit closer to the circumscribing sphere.
The shapes are often used to make dice, because dice of these shapes can be made fair. 6-sided dice are very common, but the other numbers are commonly used in role-playing games. Such dice are commonly referred to as D followed by the number of faces (d8, d20 etc.).
The tetrahedron, cube, and octahedron, are found naturally in crystal structures. The dodecahedron is combinatorially identical to the pyritohedron (in that both have twelve pentagonal faces), which is one of the possible crystal structures of pyrite. However, the pyritohedron is not a regular dodecahedron, but rather has the same symmetry as the cube.
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